Constructive Hopf's theorem: Or how to untangle closed planar curves
We consider the classification of polygons (i.e. closed polygonal paths) in which, essentially, two polygons are equivalent if one can be continuously transformed into the other without causing two adjacent edges to overlap at some moment. By a theorem of Hopf (for dimension 1, applied to polygons), this amounts to counting the winding number of the polygons. We show that a quadratic number of elementary steps suffices to transform between any two equivalent polygons. Furthermore, this sequence of elementary steps, although quadratic in number, can be described and found in linear time. In order to get our constructions, we give a direct proof of Hopf's result.
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